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\title{Loss for Classification and Regression}
\author{ML Instruction Team, Fall 2022}
\institute[]{CE Department \newline  Sharif University of Technology \newline \newline}
\date[\today]{}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Begin Your Document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
	
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\fontsize{9}{9}
\begin{frame}[noframenumbering, plain]
	\titlepage
\end{frame}

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\section{Classification Loss}
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\frame{\frametitle{Classification Loss}
	
	\begin{itemize}
		\item Both the \tc{keywords}{cross-entropy} and the \tc{keywords}{Kullback–Leibler} (KL) divergence measures the distance between two probability distributions $P$ and $Q$.
		\begin{equation*}
			\begin{aligned}
				&H(P, Q)=-\sum_x P(x) \log Q(x) \\
				&K L(P \mid Q)=\sum_x P(x) \log \frac{P(x)}{Q(x)}=H(P,Q)- H(P)
			\end{aligned}
		\end{equation*}
	
		\medskip
		\item In the above formula, $H(P)= H(P,P)$ is the entropy of the distribution $P$, which is a constant term.
		
		\medskip
		\item
		Hence, it turns out that the minimization of KL divergence is equivalent to the minimization of cross-entropy.

	\end{itemize}

}
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\frame{\frametitle{Classification Loss}
	
	\begin{itemize}
		
	\item Logarithmic loss indicates how close a distribution of prediction probability comes to the real probability distribution of the data in \tc{keywords}{binary classification}.
	\begin{equation*}
		H(p, q)=-\frac{1}{m} \sum_{i=1}^m y_i \log (p(y_i))+(1-y_i) \log (1-p(y_i))
	\end{equation*}
	
	where $y_i\in\{0,1\}$
	
	\item Cross-entropy loss indicates how close a distribution of prediction probability comes to the real probability distribution of the data in \tc{keywords}{multilabel classification}.
	
	\begin{equation*}
		H(p, q)=-\frac{1}{m} \sum_{i=1}^m\sum_{j=1}^{k} y_j^{(i)} \log (p(y_j^{(i)}))
	\end{equation*}
	
	where $y_j^{(i)}$ and $p(y_j^{(i)})$ is respectively the true and predicted probability of the class $i$-th of the sample $j$-th.
	
	
	\end{itemize}
}
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\frame{\frametitle{Classification Loss}
	\begin{figure}
		\centering
		\includegraphics[width=10cm, height=6cm]{Figs/cross_entropy.png}
		\caption{Cross Entropy Evaluation, \href{https://tinyurl.com/2mehqavm}{Source}}
		\label{fig:cross entropy evaluation}
	\end{figure}

}

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\frame{\frametitle{Regression Loss}
	\vspace{-0.5cm}
	\begin{equation*}
		\text{MSE}(y, \hat{y}) = \frac{1}{m}\sum_{i=1}^{m}(y_i-\hat{y}_i)^2
	\end{equation*}

	\begin{equation*}
		\text{MAE}(y, \hat{y}) = \frac{1}{m}\sum_{i=1}^{m}|(y_i-\hat{y}_i)|
	\end{equation*}

	\begin{equation*}
		\text{MAPE}(y, \hat{y}) = \frac{1}{m}\sum_{i=1}^{m}|\frac{(y_i-\hat{y}_i)}{y_i}|
	\end{equation*}
	
	\begin{equation*}
		\text{Logcosh}(y, \hat{y}) = \frac{1}{m}\sum_{i=1}^{m}\cosh(y_i-\hat{y}_i)
	\end{equation*}
	
	\begin{equation*}
		L_\delta(y, \hat{y})= \begin{cases}\frac{1}{2} (y-\hat{y})^2 & \text { for }|y-\hat{y}| \leq \delta \\ \delta \cdot(|(y- \hat{y})|-\frac{1}{2} \delta), & \text { otherwise }\end{cases}
	\end{equation*}

}
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\frame{\frametitle{Regression Loss}

	\begin{figure}
		\centering
		\includegraphics[width=10cm, height=6cm]{Figs/regressionlosses.png}
			\caption{Various Regression Losses, \href{https://tinyurl.com/2ldxgbc5}{Source}}
		\label{fig:regtession losses}
	\end{figure}
}


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\frametitle{Final Notes}
\centering
\vspace{50 pt}
\textbf{Thank You!}
\vspace{50pt}

\textbf{Any Question?}
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\end{document}